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Properties

Label 1-1792-1792.1445-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0857 - 0.996i$
Analytic cond. $192.577$
Root an. cond. $192.577$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.528 + 0.849i)3-s + (−0.162 − 0.986i)5-s + (−0.442 − 0.896i)9-s + (−0.683 + 0.729i)11-s + (−0.634 + 0.773i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.412 + 0.910i)19-s + (0.321 − 0.946i)23-s + (−0.946 + 0.321i)25-s + (0.995 + 0.0980i)27-s + (−0.956 − 0.290i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.812 − 0.582i)37-s + ⋯
L(s)  = 1  + (−0.528 + 0.849i)3-s + (−0.162 − 0.986i)5-s + (−0.442 − 0.896i)9-s + (−0.683 + 0.729i)11-s + (−0.634 + 0.773i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.412 + 0.910i)19-s + (0.321 − 0.946i)23-s + (−0.946 + 0.321i)25-s + (0.995 + 0.0980i)27-s + (−0.956 − 0.290i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.812 − 0.582i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0857 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0857 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0857 - 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0857 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1713116723 - 0.1572024815i\)
\(L(\frac12)\) \(\approx\) \(0.1713116723 - 0.1572024815i\)
\(L(1)\) \(\approx\) \(0.6686543273 + 0.1581722338i\)
\(L(1)\) \(\approx\) \(0.6686543273 + 0.1581722338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.528 + 0.849i)T \)
5 \( 1 + (-0.162 - 0.986i)T \)
11 \( 1 + (-0.683 + 0.729i)T \)
13 \( 1 + (-0.634 + 0.773i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (-0.412 + 0.910i)T \)
23 \( 1 + (0.321 - 0.946i)T \)
29 \( 1 + (-0.956 - 0.290i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.812 - 0.582i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (-0.471 + 0.881i)T \)
47 \( 1 + (0.991 + 0.130i)T \)
53 \( 1 + (0.729 + 0.683i)T \)
59 \( 1 + (-0.352 + 0.935i)T \)
61 \( 1 + (0.0327 - 0.999i)T \)
67 \( 1 + (0.849 + 0.528i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (-0.0654 + 0.997i)T \)
79 \( 1 + (-0.608 - 0.793i)T \)
83 \( 1 + (0.0980 + 0.995i)T \)
89 \( 1 + (-0.659 + 0.751i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.92863180359510513997354489180, −19.30704486538364978280333345043, −18.54266033412442692148058023272, −18.26262712154749105360929991534, −17.27137687579663250170601787942, −16.770803710231772710064135994870, −15.65066515492633766570428859352, −15.08640752322014979432234855055, −14.12930485251563882253904451856, −13.43383734044082206176299888147, −12.84999655951112708205674515152, −11.80177192594555295044664369576, −11.309484434845221391021107808851, −10.58370622823062724541064869384, −9.859408106347866420999594772151, −8.64336606553430513091054117603, −7.58807235877670560094791830998, −7.4133540258953528872502832408, −6.425445555441484936217591211812, −5.57196716227063244199090165648, −5.03016856017441440530229791844, −3.48507498764500649861943199806, −2.8193145370132409687407068861, −1.98751820230573953278590609749, −0.633899910310789160698385491398, 0.06888250551060297366619807374, 1.31440264109228648806634995584, 2.3649540935235876579805071180, 3.74681719049269195813872082351, 4.27899013349638283762568991692, 5.16381827314682366737276244344, 5.62354636804789273828901112017, 6.75462933757045596954469568065, 7.719141001017014302444147715332, 8.62674461563472565703999375795, 9.30528142221835282510330683784, 10.118635761928990283925896803, 10.64534521384814328792200258102, 11.735277569106527506915937225742, 12.43309327895801405877113543049, 12.74466793762284827787419685290, 14.11816276802975228467336411248, 14.77106001048921634605150647998, 15.550994784184682234937926786843, 16.247412781781749286597814818467, 16.950905079957973357598044337277, 17.22891403132652909808498710047, 18.35506060193956737314114492099, 19.12501020700015993009771787692, 20.099569974819613216070069301376

Graph of the $Z$-function along the critical line